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Function Theory 函数论


二级学科, 专业名称:函数论, 门类/类别:理学     学科/类别:数学


put function arguments in parentheses: sin(x) -- correct, sin x -- incorrect; do not omit multiplication: x*sin(x) -- correct, x sin(x) -- incorrect. x**y or pow(x,y) for power. Usual keywords are lowercase, which are different from uppercase, e.g. sin is different from Sin. Its default variable is small letter x, but its default index variable in discrete math is k.

Function Search

  • Search function with ? in the input box, e.g.

  • search function in wiki. e.g.

  • serach function in Digital Library of Mathematical Functions NIST, e.g.

    List of Math function and operations

    JavaScript math - function reference

    Classification 分类


    ▪ 4110:实变函数论 ▪ 4120:单复变函数论 ▪ 4130:多复变函数论
    ▪ 4140:函数逼近论 ▪ 4150:调和分析     ▪ 4160:复流形
    ▪ 4170:特殊函数论 ▪ 4199:函数论其他学科

    Category Functions
    constantsE, infinity, inf, e, false, i, null, pi, true, oo
    operator +, -, *, /, ^, **, >, >=, <, <=, ==, !=, <>
    complex operator add(x,y), div(x,y), mul(x,y), sub(x,y), pow(x,y), root(x,y), inv(x), neg(x)
    Algebra combine, factor, expand, inverse, lsolve, fsolve, simplify, solve, nsolve, rsolve, psolve, test, ftest, rtest
    Arithmetic abs(x), cbrt(x), ceil(x), exp(x), fix(x), floor(x), gcd(x,y), hypot(x), lcm(x,y), log(x), log10(x), ln(x), norm(x), nthRoot(x), round(x), sign(x), sgn(x), sqrt(x),
    Combinatorics bell, binomial, catalan, combination, Stirling1, Stirling2
    complexarg(x), conj(x), im(x), imag(x), re(x), real(x)
    Convert convert, toexp, tolog, totrig, totrigh, tocomplex, toint, tosum, toseries
    Construction see mathjs
    Calculus lim, limit, d, diff, int, integral, integrate, convolute, laplace, inverselaplace, fourier, semid, semiint, nthd, ode, pde, dsolve, pdsolve, odeplot, taylor, series
    Discrete math difference differences Delta sum sums sum0n sum1n sum0inf sum1inf infsum partialsum psum infsum1 partialsum1 psum1 recurrence rsolve tosum
    Expression eval, evaluate, help, ?
    Geometrycircle(1), semicircle(1), oval(2,1), disc(1), ring(1,2), area, length, tangent, slope,
    Logical and, not, or, xor, >, >=, <, <=, ==, !=, <>
    matrix see mathjs
    numeric n, nint, nsum, nsolve, see mathjs
    Probability binomial(3,x), combination(3,x), factorial(x), factorial2(x), gamma(x), random(x), P(x>0.5)
    polymonial bell(n,x), euler(n,x), binomial(n,k), bernoulli(n,x), Stirling1(n,x), Stirling2(n,x), harmonic(3,x), poly(3,x), hermite(3,x),
    number theory bell(x), binomial(n,k), bernoulli(n,x), catalan(x), euler(n,x), fac(x), factorial(x), factorial2(x), fib(x), fibonacci(x), Stirling1(n,x), Stirling2(n,x), worpitzky(x), harmonic(x), H(x), zeta(n,x), poly(3,x), hermite(3,x), fallingfactorial(n,m), risingfactorial(n,m), factorialpower(m,n), isprime, prime, nextprime, mod, powermod(m,p,n), cunningham_prime_factors, is_prime, is_prime_power, is_pseudoprime, is_pseudoprime_power, mwrank_initprimes, next_prime, next_prime_power, next_probable_prime, nth_prime, previous_prime, previous_prime_power, prime_divisors, prime_factors, prime_pi, prime_powers, prime_range, prime_to_m_part, primer, primes, primes_first_n, random_prime, twinprime,
    special see special function
    Statistics max(x), mean(x), median(x), min(x), mode(x), prod(x), std(x), sum(x), total, variance(x)
    Trigonometryacos(x), acosh, acot, acoth, acsc, acsch, asec, asech, asin, asinh, atan, atan2, atanh, cos, cosh, cot, coth, csc, csch, sec, sech, sin, sinh, tan, tanh
    Utils clone, format, has, hasnot, isinteger, isNaN, isNegative, isnumeric, isPositive, isprime, isfunction, numeric, range,
    plot draw, plot(x), parametricplot, implictplot, polarplot, tangentplot, secantplot, funplot, odeplot, complexplot(z),
    plot2D plot2D, complex2D, diff2D, integrate2D, function2D, parametric2D, contour2D, semid2D, semiint2D
    plot3Dsurface3D, plot3D, parametric3D, implicit3D, spin3D, section3D, contour3D, complex3D, function3D, wireframe3D, math3D, data3D , graph3D

    1. Math function 数学函数

      1. Standard Function 初等函数
      2. JavaScript Math function 函数
      3. Java Math function 函数

      4. Special Function 特殊函数
        B Beta beta binomial
        Cl Clausen ci chi
        dilog digamma doublefactorial
        E Ei Ein En eta erf erfc erfi
        factorial factorial2 fallingfactorial fresnelC fresnelS
        Gamma gamma gaussi
        H harmonic
        L lerch li loggamma
        P Phi pochhammer polygamma polylog psi
        si shi smallgamma

      5. Complex function 复变函数
        Basic Functions 基本初等函数, Logarithmic Functions 对数函数, Circular Functions 三角函数, Hyperbolic Functions 双曲函数, Miscellaneous Functions
        Gamma Functions 伽马函数, Bessel Functions 贝塞耳函数, Orthogonal Polynomials 正交多项式, Elliptic Integrals 椭圆积分, Elliptic Functions 椭圆函数, Hypergeometric Functions 超几何函数, Zeta Functions

      6. Inverse function 反函数
        If add prefix "inverse" to function name, it becomes to the inverse function, e.g. inversef(x) is inverse of f(x) for `f^-1(x)`. The inverse of trig function is denoted by prefix "a" onto function name, e.g. asin(x) is inverse of sin(x) for `sin^-1(x)` , other inverse function is denoted by prefix "inverse" onto function name.

      7. polymonial and number theory
        bell(x), euler(n,x), factorial(x), factorial2(x), binomial(n,k), bernoulli(n,x), Stirling1(n,x), Stirling2(n,x), worpitzky(x), harmonic(x), zeta(n,x), poly(3,x), hermite(3,x),

    2. operations

      1. Inert function:
        If add profix "s" to function name, it becomes to the inert function. Its last letter is "s" for inactivative, e.g. ds( ), integrates( ). A non-integer derivative is inert, e.g. `sin^((0.5))(x)`, to activate by removing its last letter "s" or by clicking the simplify button.

      2. to function:
        If add prefix "to" to function name, it becomes to the convert function, e.g. adding prefix "to" to sin(x) becomes tosin(x), convert x to sin.
        It is the same as convert( ) function convert to another form. e.g.
        convert exp(x) to sin

      3. Calculus function 微积分函数
        lim limit d diff ds dsolve infint int integral integrate integrates odetest semid semiint series taylor completed

        The nth order derivative function 导数
        The nthd( ) gives the nth derivative formula. It is symbolic higher derivative. The 0.5th order derivative of sin(x) is denoted by sin(0.5,x) for `sin^((0.5))(x)`. The minus order derivative is intergal, e.g. `sin^((-1))(x)` is `int` sin(x), it is different from `sin^-1(x)`, which is inverse of sin(x). A non-integer derivative is inert, e.g. `sin^((0.5))(x)`, to activate by clicking the simplify button.

      4. Integral function 积分函数
        If add profix "i" or "Integral" to function name, it becomes to the integral function. Its last letter is "i" for integral, e.g. adding profix "i" to tan become tani, tani(x) is integral of tan(x)/x.

      5. Discrete math function 离散数学
        difference differences Delta sum sums sum0n sum1n sum0inf sum1inf infsum partialsum psum infsum1 partialsum1 psum1 recurrence rsolve tosum

      6. Numeric function 数值函数 计算数学
        Its first letter is "n" for numeric, e.g. nsum( ) is numeric sum.

    3. plotting function 制图函数
      plot(x), implicitplot(x=y), parametricplot(t,t), tangentplot(x), secantplot(x), polarplot(x), overlapplot(x), complexplot(z),
      制图, 隐函数制图, 参数制图, 切线制图, 割线制图, 极坐标制图,
      surface3D(x*y), function3D(x*y),

      The following plot functions did not work on Microsoft IE and QQ browsers. Please use Google Chrome browser.
      plot3D(x*y), contour3D(x*y), wireframe3D(x*y), implicit3D(x*y*z), complex3D(x), parametric3D(t,t,t), spin3D(x), section3D(x)
      立体图, 立体外形图, 立体线框图, 立体隐函数图, 复数图, 立体参数图,

      In order to plot complex function, you should convert it to complex by tocomplex(x) before plot. Use complex(x) to set variable for complex variable in real value, and Use complex(0,x) to set variable for complex variable in imag value. Get real part by re( ) and imag part by im( ). e.g. convert log(x) to re(log(complex(x))).

    4. geometry function 几何函数
      semicircle(r), circle(r), oval(rx,ry), tangent(x)
    5. animation function 动画函数
      timer(4), sinwave(4), sunrise(4)

    6. Programing function 编程函数
      1. "is" function: Its first two letters are "is". e.g. isreal( ), isnumber( ), isconctant( )
      2. condition function : if( )
      3. repeat function : for( ) While( )
      4. block( )

    7. Internal Function 内部函数
      system function in the Java API.
    8. External Function 外部函数
      user-defined function in text .txt file, which is auto loaded. You can add your own new function.

    Add New Function 添加新函数

    Add or define new function of a single character name by f(x)=sin(x); other new function by ff(x_):=sin(x);

    Definition 定义式

    Clicking or inputing a function enters it to the Calculator, then click the definition buttion to get its definition.

    Usage 使用方法

    convert - gamma - dsolve - solve - rsolve - fsolve - lasolve - nsolve

    It is similar to math function in Java and JavaScript. See help and manual.
    You can use first function as command and "to" as ",", e.g.

  • convert `Gamma(x)` to x!
    then hit the button or the ENTER key in your keybord.


    Curve graph 曲线图

    Function Graph 函数图

    List of Functions 函数目录

    Function source 源程序

    abs activate acos acosh acot acoth acsc acsch arg asec asech asin asinh assume atan atan2 atanh

    B Beta beta binomial block

    C cbrt ceil ci circle cis Cl Clausen chi combination combine conjugate convert cos cosh csc csch csgn

    d def define definition delta diff difference digamma dilog doublefactorial ds dsolve

    E Ei Ein En evaluate eta erf erfc erfi exp expand expandexp expandtrig

    factor factorial factorialpower fallingfactorial floor For for FresnelC FresnelS

    Gamma gamma GammaQ gammaP gauss gaussi gaussian gcd gsolution

    H harmonic has hypot

    If if im infint infints infsum infsums int integral integrate integrates inverse inverseerf inversefactorial inverseGamma inverselaplace isatom isconstant iscomplex isfloat iseven isfree isfunction isinteger isnumber isodd isprime

    L laplace lasolve left Lerch lg li lim limit ln log log10 loggamma

    max min mittag mod modInverse

    n nextprime nint nlim not nsolve nsum nthd


    P Phi parametricplot partialsum plot pochhammer polar polarplot poly polyroot polygamma polylog point powermod prime prod product psi psolve psolution psum

    random re recurrence remainder replace right risingfactorial rsolve round

    S sec sech secantplot semid semiint series sgn sign si simplify sin sinc sinh smallgamma solve Sophomore sqrt sum sum0inf sum0n sum1inf sum1n swap

    tan tani tanh tangent tangentplot taylor theta tocomplex tocos tocosh todegree toerf toexp togamma tofactorial tolog toint tointegral tosum toparfrac topolar topoly topolyroot topower tosin tosinh toradian totrig totrigh totald trigamma type


    when While



  • handbook : chapter 1 function + chapter 12 special function 
    See Also